# The nature of the steady state in models of optimal growth under uncertainty

## Author

Listed:
• Tapan Mitra

()

• Luigi Montrucchio

()

• Fabio Privileggi

()

## Abstract

We study a one-sector stochastic optimal growth model with a representative agent. Utility is logarithmic and the production function is of the Cobb-Douglas form with capital exponent $\alpha$ . Production is affected by a multiplicative shock taking one of two values with positive probabilities p and 1-p. It is well known that for this economy, optimal paths converge to a unique steady state, which is an invariant distribution. We are concerned with properties of this distribution. By using the theory of Iterated Function Systems, we are able to characterize such a distribution in terms of singularity versus absolute continuity as parameters $\alpha$ and p change. We establish mutual singularity of the invariant distributions as p varies between 0 and 1 whenever $\alpha < 1/2$ . More delicate is the case $\alpha > 1/2$ . Singularity with respect to Lebesgue measure also appears for values $\alpha ,p$ such that $\alpha < p^{p}\left( 1-p\right)^{\left( 1-p\right) }$ . For $\alpha > p^{p}\left( 1-p\right) ^{\left( 1-p\right) }$ and $1/3\leq p\leq 2/3,$ Peres and Solomyak (1998) have shown that the distribution is a.e. absolutely continuous. Characterization of the invariant distribution in the remaining cases is still an open question. The entire analysis is summarized through a bifurcation diagram, drawn in terms of pairs $\left( \alpha ,p\right)$ . Copyright Springer-Verlag Berlin/Heidelberg 2003

## Suggested Citation

• Tapan Mitra & Luigi Montrucchio & Fabio Privileggi, 2003. "The nature of the steady state in models of optimal growth under uncertainty," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 23(1), pages 39-71, December.
• Handle: RePEc:spr:joecth:v:23:y:2003:i:1:p:39-71
DOI: 10.1007/s00199-002-0340-5
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File URL: http://hdl.handle.net/10.1007/s00199-002-0340-5

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## References listed on IDEAS

as
1. Futia, Carl A, 1982. "Invariant Distributions and the Limiting Behavior of Markovian Economic Models," Econometrica, Econometric Society, vol. 50(2), pages 377-408, March.
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8. L. Montrucchio & F. Privileggi, 1999. "Fractal steady states instochastic optimal control models," Annals of Operations Research, Springer, vol. 88(0), pages 183-197, January.
9. Mirman, Leonard J, 1972. "On the Existence of Steady State Measures for One Sector Growth Models with Uncertain Technology," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 13(2), pages 271-286, June.
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## Citations

Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
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Cited by:

1. Guido Cozzi & Fabio Privileggi, 2009. "The fractal nature of inequality in a fast growing world: new version," Working Papers 2009_30, Business School - Economics, University of Glasgow.
2. Santanu Roy & Itzhak Zilcha, 2012. "Stochastic growth with short-run prediction of shocks," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 51(3), pages 539-580, November.
3. Gardini, Laura & Hommes, Cars & Tramontana, Fabio & de Vilder, Robin, 2009. "Forward and backward dynamics in implicitly defined overlapping generations models," Journal of Economic Behavior & Organization, Elsevier, vol. 71(2), pages 110-129, August.
4. repec:ipg:wpaper:2014-086 is not listed on IDEAS
5. Mitra, Tapan & Privileggi, Fabio, 2005. "Cantor Type Attractors in Stochastic Growth Models," POLIS Working Papers 43, Institute of Public Policy and Public Choice - POLIS.
6. La Torre, Davide & Marsiglio, Simone & Privileggi, Fabio, 2018. "Fractal Attractors in Economic Growth Models with Random Pollution Externalities," Department of Economics and Statistics Cognetti de Martiis. Working Papers 201801, University of Turin.
7. Kamihigashi, Takashi & Stachurski, John, 2016. "Seeking ergodicity in dynamic economies," Journal of Economic Theory, Elsevier, vol. 163(C), pages 900-924.
8. Kamihigashi, Takashi, 2007. "Stochastic optimal growth with bounded or unbounded utility and with bounded or unbounded shocks," Journal of Mathematical Economics, Elsevier, vol. 43(3-4), pages 477-500, April.
9. Shilei Wang, 2015. "The Iterative Nature of a Class of Economic Dynamics," Czech Economic Review, Charles University Prague, Faculty of Social Sciences, Institute of Economic Studies, vol. 9(3), pages 155-168, December.
10. La Torre, Davide & Marsiglio, Simone & Mendivil, Franklin & Privileggi, Fabio, 2016. "Fractal Attractors and Singular Invariant Measures in Two-Sector Growth Models with Random Factor Shares," Department of Economics and Statistics Cognetti de Martiis. Working Papers 201620, University of Turin.
11. Guido Cozzi & Fabio Privileggi, 2002. "Wealth Polarization and Pulverization in Fractal Societies," ICER Working Papers - Applied Mathematics Series 39-2002, ICER - International Centre for Economic Research.
12. Takashi Kamihigashi & John Stachurski, 2014. "Interlinkage between Real Exchange rate and Current Account Behaviors: Evidence from India," Working Papers 2014-86, Department of Research, Ipag Business School.
13. Mitra, Tapan & Privileggi, Fabio, 2003. "Cantor Type Invariant Distributions in the Theory of Optimal Growth under Uncertainty," Working Papers 03-09, Cornell University, Center for Analytic Economics.
14. La Torre, Davide & Marsiglio, Simone & Privileggi, Fabio, 2011. "Fractals and Self-Similarity in Economics: the Case of a Stochastic Two-Sector Growth Model," POLIS Working Papers 157, Institute of Public Policy and Public Choice - POLIS.
15. Shilei Wang, 2012. "Iterated Function Systems with Economic Applications," Papers 1209.4849, arXiv.org.
16. Davide La Torre & Simone, Marsiglio & Mendivil, Franklin & Privileggi, Fabio, 2015. "Self-Similar Measures in Multi-Sector Endogenous Growth Models," Department of Economics and Statistics Cognetti de Martiis. Working Papers 201509, University of Turin.
17. Olson, Lars J. & Roy, Santanu, 2005. "Theory of Stochastic Optimal Economic Growth," Working Papers 28601, University of Maryland, Department of Agricultural and Resource Economics.

### Keywords

Stochastic optimal growth; Iterated Function System; Singular and absolutely continuous invariant distribution.;

### JEL classification:

• C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
• O41 - Economic Development, Innovation, Technological Change, and Growth - - Economic Growth and Aggregate Productivity - - - One, Two, and Multisector Growth Models

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